Context. The formation process of binary stars and multiple systems is poorly understood. The multiplicity rate of Class II premain-sequence stars and Class I protostars is well documented and known to be high (∼ 30% to 50% between ∼100 and 4000 AU). However, optical / near-infrared observations of Class I/Class II YSOs barely constrain the pristine properties of multiple systems, since dynamical evolution can quickly alter these properties during the protostellar phase. Aims. Here, we seek to determine the typical outcome of protostellar collapse and to constrain models of binary formation by core fragmentation during collapse, using high-resolution millimeter continuum imaging of very young (Class 0) protostars observed at the beginning of the main accretion phase. Methods. We carried out a pilot high-resolution study of 5 Class 0 objects, including 3 Taurus sources and 2 Perseus sources, using the most extended (A) configuration of the IRAM Plateau de Bure Interferometer (PdBI) at 1.3 mm. Our PdBI observations have a typical HPBW resolution ∼0.3 −0.5 and rms continuum sensitivity ∼0.1−1 mJy/beam, which allow us to probe the multiplicity of Class 0 protostars down to separations a ∼ 50 AU and circumstellar mass ratios q ∼ 0.07. Results. We detected all 5 primary Class 0 sources in the 1.3 mm dust continuum. A single component associated with the primary Class 0 object was detected in the case of the three Taurus sources, while robust evidence of secondary components was found toward the two Perseus sources: L1448-C and NGC1333-IR2A. We show that the secondary 1.3 mm continuum component detected ∼600 AU south-east of L1448-C, at a position angle close to that of the CO(2−1) jet axis traced by our data, is an outflow feature directly associated with the powerful jet driven by L1448-C. The secondary 1.3 mm continuum component detected ∼1900 AU southeast of NGC1333-IR2A may either be a genuine protostellar companion or trace the edge of an outflow cavity. Therefore, our PdBI observations revealed only wide (>1500 AU) protobinary systems and/or outflow-generated features. Conclusions. When combined with previous millimeter interferometric observations of Class 0 protostars, our pilot PdBI study tentatively suggests that the binary fraction in the ∼75−1000 AU range increases from the Class 0 to the Class I stage. It also seems to argue against purely hydrodynamic models of binary star formation. We briefly discuss possible alternative scenarios to reconcile the low multiplicity rate of Class 0 protostars on small scales with the higher binary fraction observed at later (e.g. Class I) evolutionary stages.